Numerical results for ground state and excited state properties (energies, double occupancies, and Matsubara-axis self energies) of the single-orbital Hubbard model on a two-dimensional square lattice are presented, in order to provide an assessment of our ability to compute accurate results in the thermodynamic limit. Many methods are employed, including auxiliary field quantum Monte Carlo, bare and bold-line diagrammatic Monte Carlo, method of dual fermions, density matrix embedding theory, density matrix renormalization group, dynamical cluster approximation, diffusion Monte Carlo within a fixed node approximation, unrestricted coupled cluster theory, and multireference projected Hartree-Fock. Comparison of results obtained by different methods allows for the identification of uncertainties and systematic errors. The importance of extrapolation to converged thermodynamic limit values is emphasized. Cases where agreement between different methods is obtained establish benchmark results that may be useful in the validation of new approaches and the improvement of existing methods. arXiv:1505.02290v2 [cond-mat.str-el] 15

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We report the observation of magnetic superstructure in a magnetization plateau state of SrCu 2 (BO 3 ) 2 , a frustrated quasi-two-dimensional quantum spin system. The Cu and B nuclear magnetic resonance (NMR) spectra at 35 millikelvin indicate an apparently discontinuous phase transition from uniform magnetization to a modulated superstructure near 27 tesla, above which a magnetization plateau at 1/8 of the full saturation has been observed. Comparison of the Cu NMR spectrum and the theoretical analysis of a Heisenberg spin model demonstrates the crystallization of itinerant triplets in the plateau phase within a large rhomboid unit cell (16 spins per layer) showing oscillations of the spin polarization. Thus we are now in possession of an interesting model system to study a localization transition of strongly interacting quantum particles.Published in Science 298, 395 (2002).

We study the energy and the static spin structure factor of the ground state of the spin-1/2 quantum Heisenberg antiferromagnetic model on the kagome lattice. By the iterative application of a few Lanczos steps on accurate projected fermionic wave functions and the Green's function Monte Carlo technique, we find that a gapless (algebraic) U (1) Dirac spin liquid is competitive with previously proposed gapped (topological) Z2 spin liquids. By performing a finite-size extrapolation of the ground-state energy, we obtain an energy per site E/J = −0.4365(2), which is equal, within three error bars, to the estimates given by the density-matrix renormalization group (DMRG). Our estimate is obtained for a translationally invariant system, and, therefore, does not suffer from boundary effects, like in DMRG. Moreover, on finite toric clusters at the pure variational level, our energies are lower compared to those from DMRG calculations.PACS numbers: 75.10. Jm, 75.10.Kt, 75.40.Mg, 75.50.Ee Introduction. The spin-1/2 quantum Heisenberg antiferromagnet (QHAF) on the kagome lattice provides a conducive environment to stabilize a quantum paramagnetic phase of matter down to zero temperature, [1][2][3] a fact that has been convincingly established theoretically from several studies, including exact diagonalization, 4-8 series expansion, 9,10 quantum Monte Carlo, 11and analytical techniques. 12 The question of the precise nature of the spin-liquid state of the kagome spin-1/2 QHAF has been intensely debated on the theoretical front, albeit without any definitive conclusions. Different approximate numerical techniques have claimed a variety of ground states. On the one hand, densitymatrix renormalization group (DMRG) calculations have been claimed for a fully gapped (nonchiral) Z 2 topological spin-liquid ground state that does not break any point group symmetry.13,14 On the other hand, an algebraic and fully symmetric U (1) Dirac spin liquid has been proposed as the ground state, by using projected fermionic wave functions and the variational Monte Carlo (VMC) approach. [15][16][17][18][19][20] In addition, valence bond crystals have been suggested from many other techniques. In particular, a 36-site unit cell valence-bond crystal [21][22][23] was proposed using quantum dimer models, 24-28 series expansion 29,30 and multiscale entanglement renormalization ansatz (MERA) 31 techniques. Finally, a recent coupled cluster method (CCM) suggested a q = 0 (uniform) state. 32On general theoretical grounds, the Z 2 spin liquids in two spatial dimensions are known to be stable phases, 33-35 as compared to algebraic U (1) spin liquids, which are known to be only marginally stable.36 However, explicit numerical calculations using projected wave functions have shown the U (1) Dirac spin liquid to be stable (locally and globally) with respect to dimerizing into all known valence-bond crystal phases. 15,17,18,20 Furthermore, it was shown that, within this class of Gutzwiller projected wave functions, all the fully symmetric, gapped Z 2 spin ...

By direct calculations of the spin gap in the frustrated Heisenberg model on the square lattice, with nearest-(J1) and next-nearest-neighbor (J2) super-exchange couplings, we provide a solid evidence that the spin-liquid phase in the frustrated regime 0.45 J2/J1 0.6 is gapless. Our numerical method is based on a variational wave function that is systematically improved by the application of few Lanczos steps and allows us to obtain reliable extrapolations in the thermodynamic limit. The peculiar nature of the non-magnetic state is unveiled by the existence of S = 1 gapless excitations at k = (π, 0) and (0, π). The magnetic transition can be described and interpreted by a variational state that is built from Abrikosov fermions having a Z2 gauge structure and four Dirac points in the spinon spectrum. PACS numbers: 1/L J 2 /J 1 =0.3 J 2 /J 1 =0.4 J 2 /J 1 =0.45 J 2 /J 1 =0.5 J 2 /J 1 =0.55 J 2 /J 1 =0.45, p=0 J 2 /J 1 =0.45, variance extrapolation J 2 /J 1 =0.5, p=0 J 2 /J 1 =0.5, variance extrapolation

We investigate the spin-1 2 Heisenberg model on the triangular lattice in the presence of nearestneighbor J1 and next-nearest-neighbor J2 antiferromagnetic couplings. Motivated by recent findings from density-matrix renormalization group (DMRG) claiming the existence of a gapped spin liquid with signatures of spontaneously broken lattice point group symmetry [Zhu and White, Phys. Rev. B 92, 041105 (2015); Hu, Gong, Zhu, and Sheng, Phys. Rev. B 92, 140403 (2015)], we employ the variational Monte Carlo (VMC) approach to analyze the model from an alternative perspective that considers both magnetically ordered and paramagnetic trial states. We find a quantum paramagnet in the regime 0.08 J2/J1 0.16, framed by 120°coplanar (stripe collinear) antiferromagnetic order for smaller (larger) J2/J1. By considering the optimization of spin-liquid wave functions of different gauge group and lattice point group content as derived from Abrikosov mean field theory, we obtain the gapless U (1) Dirac spin liquid as the energetically most preferable state in comparison to all symmetric or nematic gapped Z2 spin liquids so far advocated by DMRG. Moreover, by the application of few Lanczos iterations, we find the energy to be the same as the DMRG result within error-bars. To further resolve the intriguing disagreement between VMC and DMRG, we complement our methodological approach by pseudofermion functional renormalization group (PFFRG) to compare the spin structure factors for the paramagnetic regime calculated by VMC, DMRG, and PFFRG. This model promises to be an ideal test-bed for future numerical refinements in tracking the long-range correlations in frustrated magnets.

The Resonating Valence Bond (RVB) theory for two-dimensional quantum antiferromagnets is shown to be the correct paradigm for large enough "quantum frustration". This scenario, proposed long time ago but never confirmed by microscopic calculations, is very strongly supported by a new type of variational wave function, which is extremely close to the exact ground state of the J1−J2 Heisenberg model for 0.4 < ∼ J2/J1 < ∼ 0.5. This wave function is proposed to represent the generic spin-half RVB ground state in spin liquids. 75.10.Jm, 71.27.+a, 74.20.Mn The question whether a frustrated spin-half system is well described by a spin-liquid ground state (GS) -with no type of crystalline order -25 years after the first proposal [1] is still controversial, mainly because of the lack of reliable analytical or numerical solutions of model systems. For unfrustrated or weakly frustrated quantum antiferromagnets a deep understanding of the nature of the GS together with a quantitative description of the ordered phase is obtained by including Gaussian quantum fluctuations over a classical Néel state. [2,3] For sizeable frustration, instead, this description is known to break down. However, the short-range RVB state [4] does not prove a good starting point for the description of frustrated models; it rather turns out to be the exact GS of ad hoc Hamiltonians. [4][5][6] As a prototype of a realistic frustrated two-dimensional system, which has been recently realized experimentally in Li 2 VOSiO 4 compounds, [7] we investigate the spinhalf Heisenberg model with nearest (J 1 ) and next-nearest neighbor (J 2 ) superexchange couplings:on an N −site square lattice with periodic boundary conditions. In the (J 2 = 0) unfrustrated case, it is well established that the GS of the Heisenberg Hamiltonian has Néel long-range order, with a sizable value of the antiferromagnetic order parameter.[8] However, variational studies [9] have shown that disordered, long-range RVB states have energies very close to the exact one. It is therefore natural to imagine that by turning on the next-nearest neighbor interaction J 2 , the combined effect of frustration and zero-point motion may eventually melt antiferromagnetism and stabilize a non-magnetic GS of purely quantum-mechanical nature. Indeed, for 0.4 < ∼ J 2 /J 1 < ∼ 0.6 there is a general consensus on the disappearance of the Néel order towards a state whose nature is still much debated. [10] In a seminal paper, [11] Anderson proposed that a physically transparent description of a RVB state can be obtained in fermionic representation by starting from a BCS-type pairing wave function (WF), of the form

The Gutzwiller wave function for a strongly correlated model can, if supplemented with a longrange Jastrow factor, provide a proper variational description of Mott insulators, so far unavailable. We demonstrate this concept in the prototypical one-dimensional t − t ′ Hubbard model, where at half filling we reproduce all known phases, namely the ordinary Mott undimerized insulator with power-law spin correlations at small t ′ /t, the spin-gapped metal above a critical t ′ /t and small U , and the dimerized Mott insulator at large repulsion.PACS numbers: 71.10. Fd, 71.10.Pm, 71.27.+a, 71.30.+h Since Mott's original proposal [1] the correlationdriven metal-Mott insulator transition (MIT) has attracted rising interest, renewed by the discovery of novel strongly correlated materials. On the verge of becoming Mott insulators, many systems display very unusual properties, high-T c superconductivity being one spectacular example. While understanding Mott insulators and MITs is conceptually simple, calculations constitute a hard and long standing problem. Conventional electronic structure methods, such as Hartree-Fock (HF) or density functional theory in the local density approximation (LDA) cannot describe MITs, unless one allows for some kind of symmetry breaking. The standard example is long-range static magnetic order in the unrestricted HF, local-spin-density, or LDA+U approximations. This device works by effectively turning the MIT into a conventional metal-band insulator transition, thus masking the essence of the Mott phenomenon, where a charge gap appears quite independently of spin order. The fact that most known Mott insulators are indeed accompanied at low temperatures by some symmetry breaking, usually of magnetic type, further encourages the (wrong) surmise that it is not possible to describe any Mott insulator without a symmetry breaking.Another useful and popular approximation that may invite the same conclusion is based on the variational Gutzwiller wave function (GWF) and its various generalizations. [2,3,4,5] The GWF is the simplest way to improve a symmetry-unbroken, hence metallic, Slater determinant by partly projecting out the expensive doubleoccupancy charge configurations. In principle, were the projection complete, the GWF would indeed describe a Mott insulator devoid of symmetry breaking. Full projection however means zero band-energy gain, generally incorrect, except for infinite on-site repulsion. For finite projection, appropriate at finite repulsion, the GWF unfortunately describes a metallic state in any finite dimensional lattice, at least so long as the uncorrelated Slater determinant state is metallic. [3] To obtain an insulator, one is forced once again to Gutzwiller project an artificially symmetry-broken determinant wave function (WF). The main drawback of the GWF can be immediately recognized if one recalls Mott's original description of a correlation-driven insulator. Let us consider for simplicity the single-band Hubbard model at half-filling. First of all it is clear that the ...